Group for Non-linear Dynamics
Mario Natiello started the NLD group
in 1990. Currently at the Dept. of Mathematics, KTH-Stockholm.
In September 1996 the book Nonlinear Dynamics: A Two-way Trip
from Physics to Math, together with H. G. Solari and B.G. Mindlin
was published by
Institute
of Physics, UK, 1996.
Martin Zimmermann and
Sascha Firle joined in 1992 as Ph D. students.
Martin defended his thesis 6th of May 1997 and now is
in a postdoc with the Palma
of Mallorca group.
Magnus Strandaas joined in september 1994 for his "examensarbete" and in
may 1995 Björn Einarsson
started working with us on a half year programming project.
In September 1996, Jonas Fransson joined for his "examensarbete" and finished in March 1997.
Since Jan 1999 Melanie Dantzigeris working
on a "examensarbete".
Research profile
Our group is mainly interested in finite dimensional and infinite dimensional
dynamical systems. Specially
topological time series analysis,
global bifurcations leading to chaos,
spatiotemporal complex behaviour
population dynamics in biological systems.
See our recent publications and preprints.
Our current on-going projects are:
Topological organisation of strange attractors.
(Sascha and Mario, with collaboration of Solari's group).
We discuss the topological analysis of dynamical systems represented by
2-dimensional maps emphasizing the case of Poincare maps.
The central result consists in the implementation of a recent presentation of
braids as deformations of circles (Natiello et al 94) to the determination of
braid types associated with periodic orbits (up to a global torsion).
Since some braids imply
positive topological entropy, the topological analysis can be regarded
as a test of chaos.
The method is specially suited for experiments where the complete
reconstruction of the phase space for the flow cannot be achieved at a
reasonable cost.
We apply these ideas to data sets produced in a laser physics experiment
for which the reconstruction of the phase space of the flow is nearly
impossible.
Spatiotemporal structures in surface catalysis.
(Martin, Sascha and Mario, and collaboration of Eiswirth's and Baer's groups).
A 1-d partial differential equation (PDE) model of a catalysis on a
surface has been extensively studied for big cells.
The main results arise from the study
of the travelling-wave ODE (ordinary differential equation),
where a heteroclinic cycle bifurcation
(known as T-point) between two fixed points is found numerically.
This heteroclinic cycle separates the region in parameter space where
there exists 1-humped pulse solutions to either one or the other
stationary state. Also it delimits the region where spatiotemporal chaotic
behaviour of the PDE is reported. We computed the stability of both pulses
and one is found stable while the other unstable.
Homoclinic bifurcation and chaotic dynamics in a laser with injected signal.
(Martin and Mario, and collaboration of Solari's group).
A 3-d ODE model to a laser of injected signal was numerically
investigated. It has been found that the chaotic dynamics is organized by
a Shilnikov-saddle-node global bifurcation, close to the hopf-saddle-node
(type III) singularity. That it is Shilnikov
homoclinic orbit to a saddle-focus is destroyed by the collition with a
node-focus fixed point. The periodic orbit organization is studied via a
return map close to the saddle-node bifurcation.
Complex interaction between species witin a community
considering spatial temporal effects
(Sascha and Mario, in collaboration with Barbara Ekbom and Riccardo Bommarco).
We study insect movement (in particular a univoltine carabid beetle
Pterostichus cupreus ) by modeling predator-prey systems in a framework
of a random-walk individual based model, that takes spacial heterogeneties into
account. Results from experiments are translated to a computer model, which
allows us to investigate huge space and time scales. On the other hand
experience form computer models can again be described by differential equations
and thus analysed with PDE tools. The main goal is to improve the predator
efficiency in agricultural landscapes by ionvestigating the relationships
between dispersal patterns, population dynamics and individual movement stategies.
Heteroclinic-transcritical global bifurcation.
(Martin and Mario).
In the catalysis equations above, it is found that on top of the
1-loop heteroclinic cycle, there are also N-loop heteroclinic cycles
(N=2,3,4,5 reported). We study a return map model to this manifold
organization, taking into account that there is a local transcritical
bifurcation coexisting. It is found that under suitable reinjections,
there are countable many such heteroclinic cycles. With respect to the PDE
these will separate regions in parameter space where N-humped pulses to
one homogeneous state is replaced by a N-humped pulse to the other
homogeneous state, the latter being unstable in the PDE.
Thus the transition to
spatiotemporal chaos reported in these equations can be seen to arise
from the succesive replacement of stable N-humped pulses into unstable
N-humped pulses, with decreasing N, until the last 1-loop heteroclinic
cycles occurs and the turbulent regime sets in.
Homoclinic bifurcations in the hopf-saddle-node local bifurcation.
(Jonas, Martin, and Mario).
The laser equations showed that the hopf-saddle-node (HSN) local
bifurcation was acting as an organizing center of all the main bifurcations.
The Shilnikov-saddle-node appears for type III flows, while for the type I
the global reinjection provided by the laser equations, gives the possibility
of homoclinic connections to the small loop (periodic orbit) involved in
the HSN bifurcation. How the periodic orbits are organized are studied again
with a return map model.
Our scientific contacts and cooperations
Markus Eiswirth at the Fritz-Haber Institut (Berlin, Germany),
Markus Baer at the Max Planck Inst. for Physics of Complex Systems
(Dresden, Germany), and
Yannis Kevrekidis at Princeton University (USA):
Modelling and spatiotemporal structures in catalysis.
Barbara Ekbom and Ricardo Bommarco at the Sveriges Lantbruksuniversitet (Uppsala, Sweden): Movement patterns
of bugs in a predator-prey interaction.
Hernan Solari at the Buenos Aires University (Buenos Aires, Argentina): Argentinian branch of the group.
Bob Gilmore at Drexel University (Philadelphia, USA): U.S. branch of the group.
Jorge Tredicce at Nice University (Nice, France): Laser dynamics.
Are you interested?
If you are interested in knowing more about our current projects or
you want to comment, do not hesitate to
contact us...
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